A realization theorem in the context of the Schur-Szegő composition

Abstract

Every real polynomial of degree n in one variable with root −1 can be represented as the Schur-Szegő composition of n − 1 polynomials of the form (x + 1) n−1 (x + a i ), where the numbers a i are uniquely determined up to permutation. Some a i are real, and the others form complex conjugate pairs. In this note, we show that for each pair (ρ, r), where 0 ρ, r [n/2], there exists a polynomial with exactly ρ pairs of complex conjugate roots and exactly r complex conjugate pairs in the corresponding set of numbers a i .